1:
2:
(ta initials)
first name (print)
last name (print)
brock id (ab13cd)
(lab date)
Experiment 4
The ballistic pendulum
In this Experiment you will learn
• how to determine the speed of a projectile as well as the energy of the launcher that fired it
• that real-world interactions typically involve a series of energy exchanges
• the energy relationship between the linear motion of a ball and the angular motion of a pendulum
• that successful experimental results rely on careful measurement and application of technique
• to extend your data analysis capabilities with a computer-based fitting program;
• to apply different methods of error analysis to experimental results.
Prelab preparation
Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and
notes entered on these pages will be needed when you write your lab report and as reference material
during your final exam. Compile these printouts to create a lab book for the course.
To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the
content of this document and that of the following FLAP modules (www.physics.brocku.ca/PPLATO).
Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the
module in depth, then try the exit test. Check off the box when a module is completed.
FLAP PHYS 1-2: Errors and uncertainty
FLAP MATH 1-1: Arithmetic and algebra
FLAP MATH 1-2: Numbers, units and physical quantities
Webwork: the Prelab Ballistic Pendulum Test must be completed before the lab session
Important! Bring a printout of your Webwork test results and your lab schedule for review by the
!
TAs before the lab session begins. You will not be allowed to perform this Experiment unless the
required Webwork module has been completed and you are scheduled to perform the lab on that day.
Important! Be sure to have every page of this printoutsigned by a TA before you leave at the end of
!
the lab session. All your work needs to be kept for review by the instructor, if so requested.
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
26
27
In this experiment we explore the transfer and
conservation of energy and momentum in a collision of two objects. One of these objects is a small
projectile of mass m that is given a certain velocity
v by a launcher. The second object is a stationary
pendulum. As the projectile hits the pendulum,
a re-distribution of energy and momentum takes
place.
In a certain class of collisions, the projectile is
captured by the pendulum. For such inelastic collisions, there is only one combined object that is
moving at the end, and carries all of the kinetic energy K and momentum P . If the mass of the pendulum is M then the total mass of the combined
Figure 4.1: Ballistic Pendulum
object at the end of the collision is MT = M + m.
If the bob is stationary when the projectile hits, it
contributes nothing to the total kinetic energy and momentum of the system before the collision. Thus
the Law of Conservation of Momentum in this case yields:
Pbef ore = mv = Paf ter = (M + m)vT = MT vT
(4.1)
where vT is now the velocity of the combined object immediately after the collision. Knowing v and MT ,
we can use Equation 4.1 to determine vT .
As the pendulum begins to swing after the collision, another physical process takes place; a conversion
of the kinetic energy of the moving object into into gravitational potential energy as it swings up, losing
kinetic energy and gaining potential energy. At the bottom of the swing, the pendulum of mass MT has
all of its energy in the form of kinetic energy. At the top of the swing, all of the pendulum’s energy is
converted into gravitational potential energy, and as MT momentarily pauses and reverses its motion, the
kinetic energy falls to zero. Thus we can write:
1
Ebottom = K = MT vT2 = Etop = MT gh.
2
(4.2)
Combining Equations 4.1 and 4.2 to eliminate vT gives us an expression that relates the initial velocity v
of the projectile to the final elevation h of the combined object.
v=
MT p
2gh,
m
(4.3)
The length Rcm describes the radius of the arc from the pivot point to the centre of mass of combined rod,
block, and block contents. With the vertical orientation of Rcm as the base of a right-angled triangle, h
can be expressed in terms of the angle θ of the swing: R cos θ = (R − h). Equation 4.3 then becomes
v=
MT p
2gRcm (1 − cos θ)
m
(4.4)
There is another energy conversion taking place, even before the collision. In the experiment, the
launcher gives the projectile its initial kinetic energy and momentum by releasing the potential energy
and momentum of a compressed spring an converting it into the kinetic energy of the moving projectile.
Conservation of energy in this case yields
1
1
Einitial = P E = kx2 = Ef inal = K = mv 2
2
2
(4.5)
28
EXPERIMENT 4. THE BALLISTIC PENDULUM
where x is the spring displacement from its equilibrium (relaxed) length and k is the spring constant. As
the projectile is released from rest, the projectile has no initial kinetic energy. As the projectile begins to
accelerate, a conversion of energy takes place where the potential energy stored in the spring is converted
into the kinetic energy of the projectile. At the point where the projectile releases from the spring, all of
the spring’s stored potential energy has been converted into the projectile’s kinetic energy. We can then
determine a value for the spring constant k of the launcher by equating the initial potential energy of the
spring to the final kinetic energy of the projectile:
k=
mv 2
x2
(4.6)
Procedure
The launcher component of the ballistic pendulum consists of a precision spring encased in an aluminum
barrel. One end of the spring is secured to the closed end of the barrel. The other end is attached to a
piston that slides along the inside of the barrel.
The projectile rests on the face of the piston. A trigger mechanism allows the piston to be locked in
one of three force settings: short, medium or long range.
When the launcher is in the discharged position, the spring is subjected to a small compression so that
it will not rattle when released. This preload x0 is in the order of 1 mm. For this experiment, we will
assume that when the launcher is discharged the potential energy stored in the spring is approximately zero.
The pendulum consists of a rod and bob of combined mass M attached to a pivot point. When an
impact takes place, the pendulum catches the impacting mass m, changing the total mass of the pendulum
to MT = M + m, and is caused to swing about the pivot point to a maximum angle of deviation θ, relative
to the initial vertical position of θ = 0◦ .
The pendulum drags with it a pointer that stops at the limit of the swing and identifies the value of θ
on a degree scale concentric with the pivot. The pendulum then free falls back to the vertical position to
stop against the barrel.
A small amount of friction between the pointer and the scale prevents the pointer from falling back
along with the pendulum. The effect of this friction or the mass of the pointer on the system is negligible.
? The pointer, initially at rest, is accelerated along with the the pendulum arm on impact. Could it
keep moving past the limit of the pendulum arm, after the arm has stopped, and thus give inaccurate
angle readings?
Data gathering and analysis
To determine the physical characteristics of the ballistic pendulum apparatus:
1. remove the pendulum arm from the ballistic pendulum assembly by unscrewing the pivot screw.
Replace the screw for safekeeping;
2. verify that the plastic nut securing the brass weights to the bottom of the pendulum arm is tight;
3. measure with a digital scale (σ = ±0.01g) the mass of the ball m and ball/pendulum assembly MT ;
m = ............. ± ............. kg
MT = ............. ± ............. kg
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4. determine the centre of mass point of the pendulum/ball combination as shown in Figure 4.2. Make
sure that the arm of the pendulum is parallel with the length of the scale.
? The pendulum placed on the edge of the measuring apparatus will be unstable and not remain
balanced in that position. How then can you precisely determine the length Rcm ?
5. The centre-of-mass distance Rcm is the distance from the edge to the centre of the pivot hole on the
arm of the pendulum of mass MT ;
Rcm = ............. ± ............. m
Figure 4.2: Experimental setup for determining the centre-of-mass of the pendulum
6. with the launcher discharged and the ball removed, use the scale on the plunger to determine the
offset depth of the face of the piston from the front end of the barrel. You will need to subtract this
offset from all of the following depth measurements;
offset = ............. ± ............. m
7. with the ball removed, compress the piston until it latches at the short range setting. Measure the
depth from the face of the piston to the front end of the barrel. Subtract from this length the offset
depth of the piston and record the result below and as x in Table 4.2;
xshort = ............. ± ............. m
8. Repeat the above step for the medium and long range settings;
xmedium = ............. ± .............m
xlong = ............. ± .............m
9. determine the measurement error in the angle θ of the protractor scale used by the launcher;
∆θ = ±.............◦
10. and finally, replace the pendulum arm, making sure that the pivot screw is tight.
Before you begin to gather ballistic data remember to avoid sitting directly in front of the discharge path
of the launcher barrel and
CAUTION: Always wear safety glasses while using the launcher.
30
EXPERIMENT 4. THE BALLISTIC PENDULUM
1. Verify that the C-clamp securing the pendulum apparatus to the table is tight.
? What do you suppose might happen if the pendulum apparatus was discharged while not being
secured to the table? How would your measurement be affected? What physical law is at work here?
2. Swing the pendulum to 90◦ and support it in that position with one hand or someone’s assistance.
3. With the other hand, load the ballistic pendulum by placing the projectile in the barrel and pushing
it with the plunger rod until the trigger locks. This is the short range setting. Further compression
selects the medium range and finally, the long range setting.
4. Once the launcher is loaded, slowly withdraw the plunger, making sure that the trigger is indeed
locked and that the projectile has not rolled away from the face of the piston, as this would cause an
innacurate firing of the launcher.
5. Gently lower the pendulum to the vertical position, and move the angle indicator to the 0◦ mark. If
the indicator does not reach zero, you will need to subtract the offset from all your angle readings.
6. To fire the launcher, gently pull the string upward to release the trigger.
7. Perform five short range launches, recording the angle θi reached in trial i = 1 . . . 5 in the appropriate
spaces of Table 4.1. These values should be within ± 0.5◦ of one another, otherwise redo the set of
measurements.
8. Perform five trials using the medium and then the long range settings.
Have a TA check and approve your angle data before proceeding with the calculations.
!
range
θ1
θ2
θ3
θ4
θ5
hθi
∆θ
short
medium
long
Table 4.1: Experimental angle values at three force settings
• Calculate an average value hθi for the five angles θi obtained in each of the three sets of trials and
enter these in Tables 4.1 and 4.2. To avoid some lengthy standard deviation calculations, let the
error in the angle θi be the measurement error of the angle scale, ∆θ = ± 0.25◦ .
? Looking at your data, is this shortcut a valid way of estimating the error in θ?
You now need to calculate v and ∆v for the three range settings. Note that in Equation 4.4 only the
p
(1 − cos θ) term changes with θ. Chances of error will be minimized if the constant quantities are
equated to a term C and ∆C and are evaluated only once. All the angle error values must be expressed in
radians.
s
p
MT
∆m 2
∆Rcm 2
∆C
∆MT 2
C=
2gRcm
+
+
=
m
C
MT
m
2Rcm
31
...............................
......................................................
...............................
......................................................
s
∆C
C
2
− sin θ ∆θ
2 cos θ
2
p
vs = C (1 − cos θ)
∆vs = |vs |
...............................
......................................................
...............................
......................................................
+
vs = .................... ± .................. m/s
? What should be the dimensions of C? And those of ∆C/C?
• Calculate the maximum kinetic energy of the steel ball at the moment that it lost contact with the
piston of the launcher and enter the value in Table 4.2. Show a complete calculation for the short
range setting and include all other calculations as part of your Discussion.
s
2
∆vs 2
+ 2
vs
1
Ks = mvs2
2
∆Ks = |Ks |
...............................
......................................................
...............................
......................................................
∆m
m
Ks = .................... ± .................. J
v (m/s)
x (m)
x2 (m2 )
K (J)
±
±
±
±
±
medium
±
±
±
±
±
long
±
±
±
±
±
range
hθi
short
◦
Table 4.2: Parameters for the calculation of the kinetic energy K and the force constant k
If no energy is lost (or gained) during the interaction, the kinetic energy K is equal to the potential
energy V = kx2 /2 of the spring before the ball was discharged, K = kx2 /2. This is the equation of a
straight line Y = M X with Y = K, X = x2 and slope M = k/2. Plotting K as a function of x2 yields
from the slope a value for the spring constant k of the launcher spring.
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EXPERIMENT 4. THE BALLISTIC PENDULUM
• Shift focus to the Physicalab software and enter in the data window the three data pairs and corresponding errors as four space-delimited numbers: x2 K ∆K ∆x2 .
• Select scatter plot. Click Draw to generate a graph of your data. Your graphed points should
well approximate a straight line.
Unless the three points are collinear, the fit will not yield a valid result. If your partner’s calculations
!
agree with your own, then re-check the measurements of the pendulum apparatus. You need to
determine and correct the source of the error before proceeding. Consult the TA if you are stuck.
• Select fit to: y= and enter A*x+B in the fitting equation box. Click Draw to perform a linear
fit of the data. Label the axes and include a descriptive title. Click Send to: to email yourself a
copy of the graph for later inclusion in your lab report.
• Summarize the values for the slope, the Y-intercept Y(X=0) and their associated errors, then calculate
a value for k and ∆k:
slope = ............ ± ............
Y(X=0) = ............ ± ............
k = ..................... = ..................... = .....................
∆k = ..................... = ..................... = .....................
k = ............ ± ............
? The spring constant k is typically expressed in units of Newtons per metre. Using dimensional
analysis, verify that your dimensions for k obtained from the graph agree with those of N/m.
• From the slope and Y-intercept, calculate the corresponding X-intercept at Y=0:
X = ..................... = ..................... = .....................
∆X = ..................... = ..................... = .....................
X = ............ ± ............
• From this result estimate the spring preload distance x0 :
x0 = ..................... = ..................... = .....................
∆x0 = ..................... = ..................... = .....................
x0 = ............ ± ............
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
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Lab report
Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet
for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab
report submission deadline, at 11:00pm six days following your scheduled lab session. Turnitin will not
accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero.
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